How is writing for an action-only format different from one that uses dialogue? (This is in reference to writing plays)

It's simple. Action only reduces dialogue so as to focus more upon the aesthetics of stage performance and therefore evoke a more metaphorical scene or scenario. The absence of dialogue encourages more audience interpretation. In context to writing, however, there is no difference as both utilise an identical format.

Two different types of plays? Or are you referring to the action of the play, in contrast to its dialogue? I've never written plays, or studied them in terms of their construction, so I could be completely off base here.

Two different types of plays. One where the story is told through dialog and the other where the story is told through actions. Personally, I have never read or seen such a thing, there is nothing in our text book, and I can't find anything on on the Internet to help me out. I think the instructor is cracked!
So, I asked him: "I'm having a hard time answering this question. I couldn't find anything in the text book that spoke of action-only format and the Internet was no good either. Could you direct me to where I can find information that will help me answer this question?"
He said: "As the textbook states, 'Silence can be white hot. The most intense emotions are the ones you can’t express in words. When a character spews out an eloquent paragraph of anger, he is probably not as angry as if he stands, breathes hard, and turns away.'
"When speaking of action only, we are talking about pure description and explication. Dialogue adds an entirely new dimension to story writing. There is an action only movie currently in theaters called the Artist. Basically, it is predominantly a silent film where viewers must rely on action to understand the conflicts and plot of the story. Silent films were action only screenplays.
"Could you write an action only play or do you believe dialogue is necessary? Please explain."
So apparently, he was referring to screenplays, which threw me because we've only been working on theatrical plays.
As far as his re-phrased question goes, I don't think you could do an action only (silent) play. The advantage of screenplays is that you can have wide, diverse, and complex settings that help tell the story. When you act out a play on the stage your settings are much more confined and have to be more simple (although I have see some plays with pretty elaborate sets, but they were big budget). What do you all think? Have you seen a silent play? If so what was it? I'd like the chance to read something like that.

I'm with you on this one.
Theatrical plays, plays on the stage, have traditionally told their stories primarily through the vehicle of dialogue. Sure, the characters also engage in action -- they respond to one another, they may undertake action in secret that only we are privy to, all of that -- but action on the stage cannot go where action in film can. In film, we can see characters' responses far more minutely -- the twitch of an eye, the catch of the breathe -- than we can ever hope to on the stage. Film has at its disposal so many other ways to tell the story through music and imagery, and all those close-ups.
As for the original silent films, they told their stories both through broad action and inserted screens of, albeit simple, narration or dialogue. So they were not exactly dispensing with language. And where the story was visual, even with the limitations of early filmmaking, the story was still visual in a way that a play on the stage could not be. You can see in Keaton, in Lloyd, in Chaplin the emotional language of their faces and of their bodies -- magnified a hundred times on the screen -- in a way you could not on the stage. Silent plays? I think not.
There may have been a few silent films from the 20s that did not insert screenfuls of text, I don't doubt it as an artistic endeavor, but I have never seen one, and I have seen scores of silent films. In terms of modern films, there is one I can think of that is entirely without dialog. 3-Iron. It's Korean. Then there was that famous Twilight Zone episode from the 50s (with Agnes Moorehead, as a woman whose home is invaded by aliens), where there was only one line spoken at the end. But the lack of dialogue throughout was in this case the only way for the end to deliver the surprise that it does.
I find your teacher's approach eccentric. Though I should say that neither plays nor film were ever my primary fields of study.

Screen plays also have the advantage of being able to quickly change scenes. This is not impossible in a stage play, but it is difficult. The scenes have to be set up in a way that facilitates the changing of the set quickly. Furnishings and such are pretty much out of the question unless it is a one-act play set in one setting with only minor minor changes to show the passage of time and such.
But this has got me thinking. I have an assignment coming up in a week where we have to adapt one of our works (poem, short story, fable) to the stage. I was wondering if I could adapt one of my pieces silently. I do have a free verse poem that has no dialog, but it is out of the question because the action cannot be performed on stage (it involves a bike wreck), but my fable might work. Would mind looking at it and giving me you opinion on whether or not I could pull it off silently?

No, not at all. Can't promise I'll be much use, but I don't mind trying.
It occurs to me that this is what dance, particularly dance, attempts to do. Martha Grahame, Merce Cunninghame, and so on. Don't you think? Expressing through the vehicle of the body, of movement, what words perhaps could not convey.

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Well, okay, this is really not my area, adapting works to the stage, but it seems to me that if you were to stage this fable silently, you'd end up with a lot of pantomime. You can establish that the father is miserly with some opening action of some sort. How will you convey the fact that the father tells the son to wait for a year, at which point not only will the too-big shoes now fit, they will be brand new, rather than year-old used shoes? Then how will you have the farrier convey the fact that the nails the father wants are cheap? Well . . . I guess it wouldn't be difficult to have the farrier tell the father no, and to try to get him interested in another set. But the idea of low quality vs. high quality will be difficult without visual clues, won't it? The contrast would be easier with larger objects, in which the distinction in quality is more easily noted. A thin shabby dress vs. a fine dress of fine material. A cheap, slow, old bike vs. a shiny new racing bike. Don't you think?
I don't know. I think you lose a lot if you completely lose the dialogue. Any possibility the exercise is simply to get you thinking about the interaction of the two, the action and the words, and the way in which each supports (or refutes or refines) the other?

BTW, with The Nutcracker, would the story make sense to someone seeing it for the very first time, without any show notes? Traditionally, the playbill provides the story. And in fact, isn't there a voiceover narration that accompanies the dance? Maybe I'm thinking of the televised version.
I think without the summary of the story, you'd get the general idea but some of it -- the whole rat king episode, and other nuances of Clara's situation -- would be obscure.

Oh this is most definitely an exercise to get us thinking, 1) about dialog and the role it plays in telling the story on stage and 2) how actions can be used to help tell the story as well.
And I agree, it would be very difficult to convey the concept of quality nails on the stage, but what about if an introduction was included in the playbill. Not spelling the story out, but just letting the audience know the general theme of the story. Then when it got to the part where the stallion and the farrier argue about the nails, the audience would be able to presume that the argument is over the cost of the nails. Since the audience will know that the stallion is frugal they will be able to tell that he is refusing the nails the ferrier first offeres because of price.
I don't know. Our adapted work does not have to be silent, I'm just always looking for ways to challenge myself and I have a perfect score in the class so far so I figured even if I lost a few points it wouldn't hurt me.

Challenge is always good! And you should explore telling the story (or some story) silently, if that's what feels right artistically.
I think that your suggesting that the playbill include something to provide background and set context is, again, a recognition that the story cannot truly be told without the help of words. The suggestion is a good one though. If you were to stage anything silently, you would want to summarize the themes or say something about the story.
Just thinking about all of this is bound to help you integrate action and story together in more interesting ways, when you come to bring them together again. That would hold for works delivered only on the page as well. Although it is all words, some of the words narrate, summarize, gloss, and so on, while some of them directly represent the thoughts, spoken or not, of the characters.